Short answer:
Humans as individuals and as a species are really really hard to kill and then we have technology that makes the job even harder but you might have a good chance with this; I give higher civilisation no chance whatsoever.
Here's some discussion points to consider as to the viability of life on Earth in the aftermath of what you're describing. I've tried to go from the big and immediate to the smaller and more long term as best as possible:
Orbital considerations:
The mass removal is on the order of 200 billion metric tons, that will do strange things to the overall stability of Earth's rotation the first effect is probably going to be a pronounced wobble in axial tilt that will destablise the Earth's seasons in the immediate and longer term but should settle relatively quickly, on the order of decades, as it's only a surface effect. I learned all my astrophysics from ex-scientists who now write some of my favourite stories while not bad it's also not a real grounding in the subject but I'm reasonably sure that while it's a big mass it's a small enough percentage not to effect our solar orbit anything much.
Rock Effects:
There are going to be immediate pressure effects in the area where the crust is being disrupted, they'll be on a whole range of different scales and operate over many different timescales so in no particular order; there will be both plastic and brittle deformation of the underlying and surrounding lithospheric rock and there will be a much larger set of processes around the asthenospheric and tectonic response to mass removal. All these processes proceed from the fact that "nature abhors a vacuum". From sea-level the Himalaya rise nearly 9 kilometers, what most people don't realise is that mounts are much bigger underneath than on top, mountain have a "keel" of rock that holds them up and lets them float on the mantle, a bit like how an ice berg is mostly under the water line but mountains are slightly more evenly divided. The depth of the keel is determined by the weight of "over-burden" (the rock on top of it) so when you take 200 billion tonnes off the top in a week flat the keel will pop up like a cork out of a champagne bottle. This is where we get into problems as to the real effects of doing this, I can argue that even with such an unnaturally sudden release that "pop" is going to take centuries and won't be noticeable in a human lifetime and I can equally argue that there will be a massive and instantly recognisable rebound event that will cause massive destruction worldwide. I suspect that the real answer goes something like this; there are immediate and nearly explosive brittle fracture processes involving all the surfaces around the canyon due to the sudden lose of containing pressure this will proceed to a certain depth where plastic processes take over due to temperature/pressure effects (in geophysics pressure and temperature are almost interchangeable in their effects on the physical properties of rock), I'm not sure of the exact number at that location but it'll be deep, where the rock reacts in a more plastic fashion the reaction will be slower, and by definition non-explosive. Now if the process happens fast what you'll get is the whole mountain range melting due to decompression and instead of plastic or brittle rebound you just get an enormous flood basalt filling the newly formed basin. The roots of the Himalaya have another effect that will be missed when they start to be buoyed up by the removal of so much rock, they hold India back. India is moving into Eurasia at only 5 centimeters a year these days (which is still ridiculously fast for a tectonic plate) but before the Himalaya started forming it was going 3-4 times faster still, with the Himalaya basically removed and the roots no longer providing enough resistance India will lurch forward causing massive Earthquakes that you'll probably feel in London. It's also possible, depending on the exact structure and precise location and outlines of the ravine that you'll see flood basalts without decompressive melting, either because the ravine cuts too close to the asthenosphere and there's not enough pressure to prevent magmatic migration or because as India moves it opens up large scale fissures and rift volcanism occurs around the margins. This completely ignores out-gassing, the reality is going to be worse.
Atmospheric:
Any rebound event is going to be overshadowed by out-gassing from all the rocks previously held at high pressure under the Himalaya. All rock holds dissolved gas, the deeper it is the more pressure it's under and the more gas it holds, this is "usually" carbon dioxide, water and various sulfurous compounds, with the pressure off... imagine what happens to a shaken bottle of soda when you take the lid off, now imagine that instead of sugar water it's rock turning into foam across an area of two and a half million square kilometers. What does that do to the Earth's atmosphere? Nothing good, there are several effects that are important here, particulate is going to go up, and up, and up, there are going to be rock fragments, broadly similar to volcanic ash in vast quantities in every layer of the atmosphere, where the cloud is really thick you won't see the sun at all for, I don't know, probably months, maybe years. Where that material falls thickly it will bury the landscape and anything on it, call that East Asia, the North Pacific and the West Coast of North America to the Rockies. Moderate falls are still measured in feet and will destroy grass and crop lands, where it falls more thinly (there's going to be some everywhere) it'll depend on the exact chemistry of the fall (also where moderate volumes fall on forested terrain) but elemental toxicities, such as Phosphate toxicity could kill everything exposed to it. Sulfur aerosols are going to reflect and absorb sunlight in the upper atmosphere which will lower global temperatures appreciably, so you're probably looking at a few years without a summer, minimum. Longer term with some much newly exposed rock on the surface the level of atmospheric Carbon Dioxide is going to go into freefall (Carbon Dioxide is absorbed during erosion as it's washed out of the atmosphere by rain and reacts chemically with the rocks that rain falls on), this will put any surviving forests in jeopardy as trees cannot grow if the level of CO2 is too low. The resulting lack of greenhouse gasses will also create a more lasting cooling trend, possibly even resulting in an ice age. Sulfur finally washing out of the atmosphere en masse will also result in massive oceanic acidification and kill sea life on a grand scale.
TLDR:
Yeah, life will suck, for almost everything on Earth and for an extended span of years but, like I said, humans are tough, I'd give it even odds that some of us would come through even this much damage.
Please note: I'm sure I missed a couple of important thoughts, so I'll add anything else as it comes to mind.
Source: The Mediterranean Sea of America
